The Green's functions satisfy the Gor'kov equations obtained from the Hamiltonian (REF ) using the standard procedure:[1]}
\(-\frac{\partial {\cal G}( {k} ,\tau )}{\partial \tau } &=& \delta (\tau )\mathbb {1}_{4N} \nonumber \\&& + \left(\begin{array}{cc}\check{\epsilon }( {k} ) & \check{\Delta }( {k} ) \\\check{\Delta }^\dagger ( {k} ) & -\check{\tilde{\epsilon }}( {k} )\end{array}\right){\cal G}( {k} ,\tau ),\) 
